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  4. Find the number of quadrilateral which can be constructed by joining the vertices of a convex polygon of 20 sides, if none of the side of the polygon is also the side of the quadrilateral

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Q. Find the number of quadrilateral which can be constructed by joining the vertices of a convex polygon of 20 sides, if none of the side of the polygon is also the side of the quadrilateral

Permutations and Combinations

Solution:

We have number of quadrilateral
$={ }^{17} C _4-{ }^{15} C _2=\frac{17 \cdot 16 \cdot 15 \cdot 14}{1 \cdot 2 \cdot 3 \cdot 4}-\frac{15 \cdot 14}{1 \cdot 2}=(10)(17)(14)-(15)(7) $
$=2380-105=2275$
Alternatively :
$\text { Number of quadrilateral fromed }=\frac{{ }^{20} C _1 \times{ }^{15} C _3}{4}=25 \times 91=2275$